#500728

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6)(7,17)(8,14)(9,13)(10,15)(11,26)(12,19)(16,25)(18,35)(20,30)(21,29)(22,31)(23,33)(24,34)(27,28)(32,36);(1,2,4,3)(5,7,8,19)(6,12,14,17)(9,21,23,36)(10,18,24,20)(11,27,25,22)(13,32,33,29)(15,30,34,35)(16,28,26,31);(6,13)(7,18)(9,10)(11,25)(12,28)(14,33)(15,16)(17,31)(19,20)(21,27)(22,36)(23,24)(26,34)(30,35);(1,3,4,2)(6,14)(12,17)(13,33)(15,34)(16,26)(28,31)(29,32)(30,35)

Polycyclic presentation:

2^9|b^2=e,d^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407924);